SHUR IQ | Lyapunov Visualization Hub

V trajectories — real micro-drama panel

Per-brand V over W10–W17. Green: recovered within 90d. Red: did not recover. Black dashed: basin radius.

0.714

Validation score (single V, accepted)

4.41

Basin radius

3.28

Max Lyapunov exponent (Rosenstein)

Brands in panel

Animated: real W10–W17 V trajectories. Green = inside basin, red = outside.

Static reference: out/figures/V_trajectories_micro_drama_streaming.png — quadratic SDP fit, equilibrium x* = [49, 45, 61, 50, 55].

Each line is one brand's V trajectory across the eight-week panel. Green lines are brands flagged as having recovered — their V monotonically declines toward the basin floor. Red lines are brands that did not recover — their V stays elevated or drifts up. The black dashed line is basin radius 4.41; brands tracking above it are outside the sublevel set Ω.

What to read for

The visual confirmation that V decays along observed trajectories is what 0.714 looks like in motion. Green-heavy bottom half + red-heavy top half = decay-fraction signal. The pipeline doesn't just satisfy decay numerically; the curves render the convergence the metric is measuring.

The basin's robustness shows up in the ISS probe results — even random-direction shocks of magnitude 10 recover within 0–3 periods under modest mean-reversion stiffness. The basin is real, not a fitting artifact of small-sample bias.

Bimodality — two basins of attraction

K-means k=2 on last-quarter mean position. Per-cluster quadratic SDP fit recovers basin geometry the single-V fit averages over.

0.895

Cluster 0 validation (active competitor)

1.000

Cluster 1 validation (laggard)

0.923

Best-cluster decay fraction (combined)

+0.209

Lift over single-V (0.714 → 0.923)

Cluster 0 — Active competitor basin

16 brands · basin radius 3.66 · validation 0.895

Content

60.8

Narr.

59.1

Distrib.

73.1

Comm.

57.0

Mon.

66.3

ReelShort, DramaBox, Disney, JioHotstar, iQiYi, Holywater, CandyJar, GoodShort, Netflix, ShortMax, Lifetime A&E, Amazon, Google/100Zeros, GammaTime, ViU, Col-Belive

Cluster 1 — Laggard basin

5 brands · basin radius 1.00 · validation 1.000

Content

27.7

Narr.

25.6

Distrib.

28.5

Comm.

24.6

Mon.

16.5

verza-tv, rtp, klip, both-worlds-freeli, mansa

Animated: 21 brands separating into the active-competitor and laggard basins.

Composite scatter — cluster assignment

21 brands plotted on (composite, V) with cluster assignment. Horizontal dashed line: single-V basin radius (4.41). Vertical band shows the gap between the two cluster equilibria projected onto composite score.

Operational implication

For the micro-drama publish, per-cluster Lyapunov should be the default. The single-V 0.714 reading is real but pessimistic; the per-cluster fit pushes coverage to 0.923 by routing each holdout transition to its lowest-V cluster. Reporting both readings is the right move — single-V for cross-vertical comparison, per-cluster for within-vertical structure.

Lyapunov-adjusted stack ranking — W17

Composite vs. adjusted side by side. Bar shows adjusted composite. Arrow shows rank delta.

ReelShort — overtakes DramaBox on stability tiebreak

−2

Netflix — V 4.49, outside basin

−4

Google/100Zeros — V 10.13, worst structural fragility

GoodShort — biggest stability-driven climb

Animated: 21 brand-rank lines morphing from composite ranking to Lyapunov-adjusted ranking.

Hover any row for full V reading and stability class. Bar fill color reflects stability: blue stable, warm marginal, red unstable.

Capital efficiency — per-brand intervention scoring

Same intervention applied across the V gradient produces different ΔV. Capital efficiency = −ΔV / cost in millions.

Archetypal interventions scored per brand

7×

Spread on community/fandom across V gradient

$1.2M

Cost of top-ranked intervention for high-V brands

1.694

Best capital efficiency in panel (google-100zeros)

Animated: a single +6 community-strength intervention applied at three V positions. The same dollar buys more ΔV at higher V.

Static reference: out/figures/capital_efficiency_micro_drama_streaming.png — per-brand ranked interventions for shortmax (near equilibrium), lifetime-ae (marginal), google-100zeros (far from basin).

Reading the chart left-to-right shows ten archetypal interventions ranked by capital efficiency for each of three sample brands. The same intervention category appears at different ranks for different brands — that's the V gradient at work. A community / fandom program is mid-rank for shortmax (V 0.33, near equilibrium) and top-ranked for both lifetime-ae (V 2.44) and google-100zeros (V 10.13). The dollar cost is identical; the ΔV is not.

Why this matters for client recommendations

Per-brand intervention rankings are not transferable across brands at different positions in the basin. A “best practice” for a near-equilibrium brand may be capital-inefficient for a far-from-basin brand, and vice versa. The Lyapunov layer turns this into a quantitative reading rather than a qualitative judgment call.

The next sprint replaces the discrete intervention catalog with a control-Lyapunov-function (CLF) layer that optimizes Δx against a budget constraint as an LP/QP, scoring the continuous space rather than ten archetypes. The capital-efficiency surface seen here is the empirical rationale.

ISS / disturbance recovery

Random-direction shock injection, magnitude {2, 5, 10}. Recovery time = periods until V drops back below pre-shock level.

Median recovery (all magnitudes, accepted V)

Max recovery, large shock (~‖dx‖=10)

105

Probes per magnitude, micro-drama

2.58

Median peak V after large shock

Source: out/figures/iss_recovery.png. Bars by industry × magnitude. Micro-drama uses the accepted V; viral_short_form and subscription_streaming use fallback V's whose absolute scale is not comparable — read those as relative-shape comparators.

For the accepted micro-drama V, even large-magnitude shocks recover in 0–3 periods under the standard mean-reversion forward model. The basin is robust to disturbance — brands inside Ω revert quickly even after large random-direction kicks. This is the operational meaning of input-to-state stability for the brand-intelligence application: small or moderate shocks (competitor moves, platform changes, cultural events) get absorbed by the dynamics rather than propagating.

Caveat on comparator readings

Viral short-form and subscription streaming were rejected at validation; their ISS bars use a fallback V whose absolute scale is not comparable to the accepted micro-drama V. Read the comparator bars as relative-shape signals, not absolute recovery times. The next sprint replaces the generic mean-reversion forward model with a fitted dynamics model per industry, which would let ISS distinguish soft vs. brittle stability quantitatively.